> ... If you have a set defined by {2,> 3, 6} the sum of its elements is 11. If you have a set defined by> 9/10^n for all n in the set of whole numbers not including zero, if> you summed

But how do you sum an infinite set? This is your dilemma. You donot want to use the standard definition which involves a limit.So you need to replace this standard definition with something.You have an informal understanding of the sum of an infiniteset "keep adding stuff until you are finished",however, you cannot define "until you are finished" foran infinite set.

> all of its elements you would get 0.999...

Agreed. 0.999... is the sum of the infinite set9/10^n for all n in the set of whole numbers not including zero.We know what the sum of a finite set is.What is the sum of an infinite set?